Page 15 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes From (ii) and (iii) we have
F(x h) F(x) 1 x h 1 x h
f(x) = f(t)dt f(x)dt
h h x h x

1 x h 1 x h
= f(t) f(x) dt f(t) f(x) dt
h x h x

F(x h) F(x) 1 x h
Lim f(x) Lim f(t) f(x) dt 0 [Using (i)]
h o h h o h x
F(x h) F(x)
or Lim f(x) 0 … (iv)
h o h

Since modulus of any quantity is always positive, therefore

F(x h) F(x)
Lim f(x) 0 … (v)
h o h
Combining (iv) and (v), we obtain
F(x h) F(x)
Lim f(x) = 0
h o h

F(x h) F(x)
Lim = f (x)
h o h
F (x) = f (x).
Theorem: Every point of continuity of an integrable function f (t) is a Lebesgue point of f (t).

Proof: Let f (t) be integrable over the closed interval [a, b] and let f (t) be continuous at the
point x .
o
f (t) is continuous at t = x implies that > 0, a > 0 such that,
o
|f (t) – f (x ) | < , whenever |t – x | < .
o o
x o h x o h
|f (t) f(x )|dt dt h whenever|h| .
o
x o x o
1 x o h
|f (t) f(x )|dt … (i)
o
h x o
Now h 0 0. So from (i), we have
1 x o h
Lim |f (t) f(x )|dt 0 … (ii)
h o h x o o

1 x o h
Now Lim |f (t) f(x )|dt
o
h o h
x o
1 x o h
Lim |f (t) f(x )|dt 0 [Using (ii)]
o
h o h x o

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